Extending the time line by infinitesimals

• Concurrent event cascades:
  • oversimplification
  • real world systems
    • → small delay times for message transfer or state change
    • → lots of additional (unknown) parameters
  • modeling
    • include delays (cures cascade problems)
    • use abstract "infinitesimal" values
• Set of hyperreal numbers ^*ℝ:
  • totally ordered field
  • includes ℝ
  • includes infinitesimals
    • ε > 0, but smaller than any real positive number
    • large set (2ε, −ε, ε²,..) around 0
  • includes unlimited numbers
    • ω := 1/ε
    • large set (2ω, −ω, ω²)
  • can be explicitely constructed (as a set)
    • uses advanced set theory and logic (axiom of choice, ultrafilters)
  • allows to simplify analysis (nonstandard analysis)
• Hyperreals in the computer:
  • only a finite subset (as for reals)
  • sufficient for our purposes
    • pair (t,r) of floating point numbers t, r
    • represents t + rε